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REPORT SAMSO- TR-75-297 {SUPERSEDES TR-73-220) !> - I I < Q <£ Titan IIIC Guidance with the Carousel VB Inertial Guidance System T. E. DARNAUD and J.B.SHAUL Guidance and Control Division Engineering Science Operations The Aerospace Corporation El Segundo, Calif. 90245 \ i 15 July 1975 Prepared for SPACE AND MISSILE SYSTEMS ORGANIZATION AIR FORCE SYSTEMS COMMAND Los Angeles Air Force Station P.0. Box 92960, Worldway Postal Center Los Angeles, Calif. 90009 Approved 1 / /J. R. Allder, Director Guidance and Flight Dynamics Subdivision Guidance and Control Division Engineering Science Operations C j). D7 Baxter, Director Titan III Directorate Vehicle Systems Division Systems Engineering Operations Publication of this report does not constitute Air Force approval of the report's findings or conclusions. It is published only for the exchange and stimulation of ideas. ^RICHARD E. WOLFSBERGER, Cornel, USAF Assistant Program Director Expendable Launch Vehicles SPO This report supersedes and replaces TR-0073(34 1 3 -02)- 1 , dated 29 November 1972. UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE fJWi*. Data Bntatad) REPORT DOCUMENTATION PAGE SA MS0.¥TR - 75-297 READ INSTRUCTIONS BEFORE COMPLETING FORM *• COvT ACCESSION NO. S. RECIPIENT'S CATALOG NUMBER 5. TYPe OP REPORT A PERIOD COVEREO 7. AUTMORfsJ ) T. E. yfearnau dRMB J. RT^haul) PggTTW: ACT OR GRANT NUMBERf.J 9. PERFORMING ORGANIZATION NAME AND ADDRESS The Aerospace Corporation El Segundo, Calif. 90245 10. PROGRAM ELEMENT. PROJECT. TASK AREA * WORK UNIT NUMBERS !S 11. CONTROLLING OFFICE NAME AND ADDRESS Space and Missile Systems Organization Los Angeles Air Force Station — ■ Los Angeles, Calif. 90045 £ 14. MONITORING AGENCY NAME A AODRESS (II dt Hat ant I mm Control I Inf Ollleo) IS. SECURITY CLASS, (ol Ihlo tapoH) Unclassified IS*. DECLASSIFICATION/DOWNGRADING SCHEDULE 16. distribution statement (oi thio Raport) Approved for public release; distribution unlimited 17. DISTRIBUTION STATEMENT (ol Iho abotracl antarad In Block 30, II dl liar an I from Ha port) >9. KEY WOROS ( Continue on nvirat «id« If nKMitry and Identity by bloc* nunbar) Guidance Equations Inertial Navigation ~ Carousel VB -3 , Rocket Steering Law \ Vehicle-Borne Computer Program BSTRACT (Contlnua on ravmraa alda It ttacaaaary and tdantlty by block numb«0 The Titan II^C Standard Space Launch Vehicle, starting with Vehicle 26, will use the Carousel VB Inertial Guidance System for navigation, guidance, and digital flight controls. ''This guidance system consists of a Carousel VB Inertial Measurement Unit (1MU) and a MAGIC 352 Missile Guidance Com- puter (MGC), both manufactured by Delco Electronics, General Motors Corporation, The C-VB IMU is a modification of the C-IVB inertial navi- gator currently in airline service as a primary navigation system. — ^ 00 ruKM 1473 irACSiMlLCl a UNCLASSIFIED SECURITY CL AMI F 1C AT ION OF THIS RAOC !«*•« DM* «**•»•« SB®®®*** SECURITY CLASSIFICATION or THIS RA<3E(Trh«n Dim Bill) ABSTRACT (Conttnumd) ' A unique feature of the Carousel IMU is that two of the three gyro/ accelerometer sets are revolved at 1 rpm with respect to the stable platform. The effect is to partially cancel certain instrument errors associated with the revolving instruments. Conversion of this instrument from commercial airline navigation service to guidance and control of the rocket boost vehicle presented a guidance software design task that is the subject of this paper. UNCLASSIFIED SECURITY CLASSIFICATION OF TNI* RAOCfOM* Oat* PREFACE The guidance equations derived in this paper were developed by the authors in a joint effort with D. L. Kleinbub and A. C. Liang of The Aerospace Corporation. A summary of the first issue of this report was presented at the Sixth Hawaii International Conference on System Sciences January 11, 1973 and was published in the proceedings. CONTENTS m :\ m m fM :ni poduction MISSION SUMMARY AND VEHICLE DESCRIPTION General Vehicle Description . . . . 1 . Stage 0 2. Core Stages I and II Rage III Control Module Mission Description (Flight Plan VIU 1 . Ascent to Parking Orbit 2. Parking Orbit 3. Stage HI First Burn 4. Transfer Orbit 5. Final Orbit Injection 6. Paylcad Separation GUIDANCE, STEERING, AND NAVIGATION Gene ral Guidance Thrust Act deration Prediction Pitch Stee ring Yaw Steering Estimate of Time to Go 5. Integral Control . . . . 6. Maneuvering Equations Navigation Accelerometer Resolution and Compensation 2. Gyro Drift Compensation f PRECEDING PIGS BLANK-HOT FUME] ' T* - 3 - CONTENTS (Continued) 3. Transformation of Velocity to Inertia] x, y, 7, Coordinate System 4. Vehicle Inertial Position . 5. Gravity Computations 6. Vehicle Inertial Velocity D. Attitude Errors . 1 . Synchros 2. Attitude Error Computations REFERENCES FIGURES 1. Titan I1IC External Prolile . ... . 2. Typical Titan IIIC Mission Profile 3. Carousel VB IMU Geometry 4 Navigation Block Diagram TABLE 1 . Carousel VB IGS Characteristics . - SECTION I INTRODUCTION Referent el is a complete specification of the guidance equations for the Titan HIC launch vehicle. It includes all information necessary for programming the launch vehicle guidance computer. Reference 1 does not, however, include any discussion of the guidance algorithms or the derivation of equations. The equations specified by Ref. 1 and discussed in this document are applicable to SSLV C-26 and subsequent vehicles. Section II, a very general description of a typical Flight Plan VII mission and of the Titan IIIC vehicle, is included as an aid to understanding the inflight equations, some aspects of which arc vehicle- and/or rnission- peculia r. Section III is a discussion of the Titan IIIC navigation, guidance, and steering. - 5 - SECTION II MISSION SUMMARY AND VEHICLE DESCRIPTION A. GENERAL The Flight Plan VII mission consists of injecting a payload(s) into a synchronous or near- synchronous equatorial orbit using the Titan IIIC SSLV. The combined vehicle Stages II and III, together with the payload, are injected directly into an e'liptical parking orbit having a perigee of approxi- mately 80 nmi and an apogee of approximately 235 nmi. At the first descending or ascending node, depending on mission options, a first burn of Stage III produces an elliptical orbit with an apogee of approximately 19,323 nmi and an orbital inclination . educed by approximately 2-1/4 deg. At apogee, a second Stage III burn produces the desired synchronous (or near synchronous) equatorial orbit. B ■ VEHICLE DESCRIPTION The Titan IIIC launch vehicle shown in Fig. 1 consists of a solid motor stage and three liquid engine stages plus a control module. The solid motor stage is referred to as Stage 0, and the three liquid engine stages are referred to as Core Stages I c II, and Stage III. The control module and Stage III are sometimes collecti vely called the transtage. The stages are briefly^ described in the following paragraphs. 1. STAGE 0 Stage 0 consists of two solid propellant rocket motors positioned parallel to the standard core in the yaw plane. Thrust vector control is accomplished by injecting oxidizer (N^O^), pressurized by gaseous nitrogen (N^), into any of four quadrants of the nozzles. PRECuDINO PAJS BLANK-HOT PIIirsB" " - 7 - 1 PAYLOAD FAIRING- STATION FAIRING AND PAYLOAD- INTERFACE STAGE il AND STAGE III- SEPAR ATION PLANE -CONTROL MODULE STAGE II STAGE I A.'O STAGE II- SEPAR ATION ’’LANE /' 7 °h ' l\ ]/ — 24. 00 R PK\ “ ' - 360 - 15 | \ V-,z 125.80 (typ each segment) STAGE I — -12.00 R STAGE 0 (2 SRMs)- 5 SEGMENT SOLID- 6 OFFSET v r T "1 14- -TVC TANK H \ 2° OFF SET — S^O’ GIMBAL-Jf-^n 5° 3 O' GIMBAL p (max) —1213. 10 —1224. 3 1 1240.68 — 1274. 12 1304.25 —1326.00 —1348. 10 (nozzle exit) - 1375.79 ROLL AXIS HEATSHIELD —r'/ b.L.O. THRUST CHAMBER—' | COVERS ! -117° 24' (typ) 85.00 R- (typ) <fc. YAW AXIS^jargex 126. 51 (typ) I -W. L. 60 -<E PITCH AXIS Figure 1. Titan IIIC External Profile - 8 - z. CORE STAGES I AND II Stages I and II are powered, respectively, by dual and single *hrust chamber, turbopump-fed, liquid propulsion systems. These systems utilize storable propellants, a 50:50 mixture of hydrazine and unsymmetrical dimethylhydrazine (UDMH) for fuel and nitrogen tetroxide (N^O^) for oxidizer. 3. STAGE III The Stage III propulsion system consists of two pressure-fed engines. Stage III also contains an attitude control system (ACS), which provides atti- tude control during coast, ullage control prior to main engine burns, and velocity additions for vernier phases and for satellite ejections. The pro- pellants used for the main propulsion system are the same as for Stages I and II. The attitude control system employs hydrazine monopropellant engines. 4. CONTROL MODULE The control module contains the major elements of the flight control, inertial guidance, electrical, telemetry, tracking, and flight safety systems. The control module is attached to the forward end of Stage III and remains attached throughout flight. C. MISSION DESCRIPTION (FLIGHT PLAN VII) The following is a brief mission description of a typical Flight Plan VII. All values given are only approximate, and any discussion of the Flight Plan that does not contribute to understanding the guidance equations has been intentionally omitted. A complete description of a Flight Plan VII mission is given in Refs. 2 and 3. Figure 2 is an illustration of the Flight Plan. 1. ASCENT TO PARKING ORBIT The vehicle is launched vertically from Air Force Eastern Test Range (ETR) Pad 41 or 40; shortly after liftoff, it is rolled from a pad azimuth of 100. 2 deg to a flight azimuth of 93 deg. Following the roll maneuver, an open-loop pitchover, with load relief during the region of max Q, is performed for the remainder of Stage 0 burn. Stages I and II are closed-loop guided; - 9 - ss •H the engines burn for approximately 149 and 207 sec, respectively, and the stages burn to propellant exhaustion. At Stage II propellant depletion, the vehicle is nominally in an elliptical orbit 80 X 235 nmi. Further, in the case of a low performing vehicle, the minimum orbit is 80 X 95 nmi. 2. PARKING ORBIT Immediately after injection into the parking orbit, the vehicle attitude is adjusted to the standard orbital orientation: vehicle longitudinal axis normal to the geocentric radius vector, the nose of the vehicle pointed approxi- mately along the velocity vector, and the vehicle pitch plane coincident with the orbital plane. After coasting in the parking orbit to near the f-rst descending or ascending node, the vehicle is reoriented in yaw as a startup attitude for the first burn of the transtage. Concurrently, a roll maneuver is performed so that the vehicle telemetry antenna is pointed toward the desired ground tracking station upon completion of the reorientation maneuver. 3. STAGE III FIRST BURN Close to the equatorial crossing, the Stage III engines are ignited and burn for approximately 300 sec. The vehicle is injected into an elliptical orbit with an apogee of 19, 323 nmi. The first burn includes an orbital plane change maneuver of approximately 2. 25 deg. If required, the ACS of Stage III can adjust the orbital parameters by a vernier velocity addition. Nominally, shutdown of the Stage III engines is controlled so that a 6- sec ACS vernier phase is required. 4. TRANSFER ORBIT During the transfer orbit, the transtage performs certain maneuvers to meet thermal control requirements. Currently, three options axe being considered for thermal maneuver. The first, called rotisserie, is an oscilla- tory roll maneuver of approximately ±115 deg at a rate of 1 deg/sec, with a - 11 - f -min dwell time at cat. h cxlremo position. In the sei ond maneuver, toasting. th< vehicle is simply turned hack and forth in space at widely- spaced time intervals, l’he third is a continuous roll maneuver between 1 and 2 deg/ set. In all three options, the relationship of the sun vector to the vehicle or spacecraft axes is specified by the payload thermal requirements. In addition to the thermal maneuvers, the vehicle is oriented several times during the transfer orbit tc an attitude that permits reliable reception of telemetry. Finally, shortly before reignition of the Stage III engines, the vohic le is reoriented to a startup attitude for the second burn, which also points the telemetry antenna earthward. n. FINAL CRB IT INJECTION The second Stage III burn, of approximately 104- sec duration, injects the payload and /ehicle into a circular orbit at 19,323-nmi attitude, with a near zero-deg inclination. 6. PAYLOAD SEPARATION After the second Stage III shutdown, the vehicle is reoriented for the satellite separation phase. This orientation can vary depending on the selected mission. After sufficient time for stabilization at the desired atti- tude, the ACS is switched (o the payload release coast mode; after sufficient time for stabilization in this mode, the payloaa is released. At this point, the Titan IIIC mission can be ended with a transtage shutdown sequence. Alternatively, the equations provide an option for performing a short propellant settling burn, another payload release sequence, and, finally, the transtage shutdown sequence. In both of these payload release options, the equations issue a variety of required discretes and can perform multiple reorientations as specified by payload requirements. - 12 - SECTION III GUIDANCE, STEERING, AND NAVIGATION A. GENERAL The vehicleborne digital computer program is divided into three distinct sections: digital flight control equations, ground equations, and guidance equations. This document treats only the guidance equations; the other programs are mentioned only when they interface with the guidance equations. The carousel sensor and gimbal geometry is given in Fig. 3, while principal computer and sensor characteristics are tabulated in Table 1. Figure 3 and Table 1 are modifications of data taken from Ref. 9. The guidance equations are organized into various subprograms. A main control or executive program controls logical routing to appropriate calculations , depending on the phase of flight and/or significant flight events. The navigation equations accept inertial measurement unit (IMU) data and compute current vehicle position and velocity. Booster steering equations calculate open-loop steering for Stage 0. Powered flight equations provide vehicle steering as a function of navigational quantities and desired end con- ditions. Several coast phase subprograms provide computations for desired vehicle attitudes and event initiations during nonpowered flight. Finally, the specific flight plan and the multistage Titan 1IIC vehicle necessitate numerous subprograms to repeatedly reinitialize the guidance equations for each phase of flight and to issue discretes to control events such as engine shutdown, engine ignition, and stage separation. The following paragraphs describe the guidance philosophy and overall logic flow of the guidance equations. B. GUIDANCE The guidance philosophy governing the guidance equations presented in this document is commonly termed 'explicit" guidance. Generally, an - 13 - v> H D. or U tO m D W Ul .. I -J v> a o a uj z o Ui > o o o u. < h- z UJ UJ - I £ h* O _J UJ -J H JO X < 0 or v> 1 5 5 K ? = £ I U J~ t o x < i < >- J2 X < * < >- ^ *5 ‘5 s *3 g Oi u o V? UJ UJ UJ UJ UJ -J -J -J -J u X u X o X u X o X UJ UJ UJ UJ UJ >> > > > > > ft cr O •- t-P ^ >* o td o £ o a D 2 CQ > u* X UJ H Z UJ a O o z o a: x ¥ ° ? * >" UJ ” N x“>- o z < cn 05 ^ UJ u a: < t/> V) UJ U z t/> J- w w a o *- z z < to z < — ffl ui o S UJ o x ~J ? h a a h -J o < D UI i/> <0 _l ^ f U. UJ - O Z J *- uj O x I O -j z o J— i= < s u u_ h J < J 1 . t- ^ ,J < ►- ,? It w y a or § — Q. O' a: < ui t~ « UJ z 0 w £ < UI f- a: 2 -r Q lit O UI y 2 O X Z D 5 ~ K t- < Q O 2 i < • • >- or < a: h m a < a z o <r >- o o z < (X UJ a UJ 0 u < UJ 1 UJ (X o o z < vi < w 5 x < « a; 0) d o aJ O CO <b U r-* && ■ H h - 14 - explicit guidance scheme is one in which an explicit solution to the powered flight dynamics is computed in flight and the desired end result of guidam e is explicitly defined in the guidance equations. The Flight Plan VII guid- ance equations, deoenclmg upon the flight phase, define different desired end conditions. The burns of Core Stages I and II are planned for a specific position magnitude, velocity magnitude, radial velocity, and preplanned orbit planes. The end result of guidance for the first burn of Stage III (transtage) is injection onto a transfer ellipse whose apogee, semi-major axis orientation, and orbital plane are specified. Finally, during the second burn, the transtage guides to a near-circular orbit at the transfer ellipse apogee radius and the orbit piano is specified (approximately equatorial). References 4, 5, and 6 all contain general discussions of explicit guidance algorithms. 1. THRUST ACCELERATION PREDICTION A basic requirement of the guidance equations presented here is a prediction of vehicle performam c, particularly of thrust acceleration. For this report, "thrust acceleration" is used as an abbreviation for "accelera- tion (of the vehicle) due to thrust. " The following paragraphs derive a time-dependent expression for thrust acceleration. Consider the variable T , defined as r = W/W (1) whe re W - total weight of vehicle at any time W • rate of change of vehicle weight T^ can be interpreted as the time remaining to zero vehicle weight (mass intercept), if a constant W is assumed. - 16 - Since specific impulse is 1 = Thrust/ W sp and thrust acceleration is A th -- Thrust/ (W/g) by solving for thrust and weight as ( 2 ) ( 3 ) W = T W Thrust = W1 sp ( 4 ) one can express A„,, as 1 H A—u - gl /T TH 6 sp r ( 5 ) Further, since the effective exhaust velocity is given as V = gl e 6 sp ( 6 ) then it follows that A TH = V e /T r ( 7 ) Finally, since T^ is a linear function of time, then A^,^ as a function of time t can be expressed as A TH' 1 ' - V e /(T rO* t) ( 8 ) where T r Q = value of T r computed at t = 0. - 17 - 2. PITCH STEERING I" polar coordinates , the two-body equations of motion for a vehicle in a central for. c field gives the radial acceleration as: ^ ^THr ” wher< ^THr ~ rdciia l thrust acceleration L. n w *'■ centrifugal acceleration 2 |i / R ~ acceleration due to gravity If a time-dependent guidance steering law for the radial thrust pointing d 1 rection is assumed to be A THr /A TH T A ' Bli f WR 2 - ci 2 R)/A th ] then, upon substituting for A^, one finds that Eq. (9) becomes RMA-t Bt >A TH (11) This steering law, the familiar "linear sine" law, has been applied to numerous guidance programs. It has the advantages of simplicity and very near fuel optimality for current ro''k of - r '-.'>'.v c 'n c ,i u. Since explicit guidance equations require a prediction of vehicle motion, it is desirable to time-integrate Eq. (1 1} twice to estimate future vehicle radial velocity and position. However, before these integrations are attempted, Eq. (11) can be simplified to reduce integration complexities. - 18 - The* linear assumption of T can be written as t ( 12 ; Substituting Eq. (12) into Eq. (11) yields R = (A + BT r0 - BT r )(A TH ) (13: And, after some algebraic manipulation, R = -BT A th + (A + BT rQ )(A TH ) = -BV + (A i BT r ) (A_ u ) e rO TH (14) In Eq. (14), the quantities A,B,V o , and are assumed constant; therefore, Eq. (14) can be rewritten as R = A' + B A r r T H (1 5> Time integration of Eq. (15) from a given time (t = 0) to cutoff (t - T ) yields V rr V r T r T + / 8 A' dt + B / 8 A„ u dt r Jn r J 0 TH ( 16 ) /•T ft f, % *> R, = R + V T + f r g dt + B *fn 8 f A TH dS JO Jo dt - 19 - where &S* ■' s M 4 WM |g|| il V = present radial velocity r 1 V r - radial velocity at cutoff rf iir %k4. m R = present ra fiai position = radial position at cutoff The integrals of thrust acceleration in Eq. (16) can be evaluated using the time-dependent expression for A^^, given in Eq. ( 1 Z) : nil h M r / s Jo A.dt - V Th g A ds di - V T - V T 4 V T IH g g g rO eg in these integrals, it is assumed that V is known. A derivation of the b g second integral (a^) can be found in practically any discussion of an explicit guidance technique, such as R .. 4. To solve for the steering coefficients A^ and B f> one must substitute the values for the integrals in Eq. (17) into Eos. (16): V c - V 4 A' T + B V rf r r g r g R, R + V T 4 A' (T 2 /2) 4 B (a, ,) I r g r g r 1 2 it (19) Solving for yields B = R, - R - (T / 2)( V , + V ) r f g ” rf r T ro - <y 2 > + V T e g and for A', r A' = (1/T )(V , - V - V B ) r g rf r g r ( 20 ) Finally, the expression for the desired radial acceleration A^ .s obtained by substitution into Eq. (15): A r R = (1 /T )(V , - V g rf r - V B ) + B A, g r' r r H ( 21 ) 3. YAW STEERING The yaw steering is derived in a manner analogous to the pitch steering scheme. In the yaw case, a desired orbital plane is specified uy a vector normal to the plane. Normal error parameters R and can be computed as the dot product of vehicle inertial position and velocity with the specified normal vector. Since the desired value of both error parameters is zero at cutoff, the following equations can be used to determine the yaw steering coefficients A and B : f n n 0 = V + A T + B V n n g n g 0 = R + V T + A T / 2 + B a, _ n n g n g n 1 2 ( 22 ) - 21 - 4. ESTIMATE. OF TIME TO C.O * .1 The estimate of time to go is derived from AM, the vehicle angular momentum to bo gained, which is given as AM = M f - M "M where r<; . « M^ = desired vehicle angular momentum magnitude at cutoff M - present vehicle angular momentum magnitude Angular momentum to be gained is also given by the integral equation AM = where T ^ is the time until rocket engine cutoff. To estimate time to go, solve the integral Eq. (24) for T by a 8 numerical integration technique. The magnitude of angular momentum is given as M = Rv v^ - velocity in the tangential direction R = position magnitude Per unit mass Differentiating Eq. (25) produces M = Rv t + Rv^ (26) Another expression based on orbital and tangential thrust acceleration is \ = -\ R/R + A THt (27) where is the vehicle thrust acceleration magnitude in the tangential direction. Substituting the expression for into Eq. (26) yields for M the simple expression M = Rv t + R(-v t R/R + A THt ) = RA THt (28) The integral of Eq. (24) is evaluated using Simpson's integration formula, with the integrand computed at t = 0, t = T , and three equally- 6 spaced points in between. The first factor in the integrand, position magnitude, is computed at the five desired time points by a linear interpolation between the present vehicle position magnitude R and the final position magnitude R^; thus, R(tj) = R + (j - l)(R f - R)/4 j = i, ...5 (29) Thrust acceleration magnitude as a function of time has been derived m Paragraph III. B. 1. Thus, it follows that a th<V ^ V J ' T r0- - J )T / 4 g j = 1. (30) Where T is tnc time-to-go from the last major cycle decremented by 1 sec g - 23 - The tangential acceleration A,p^ is obtained from a square root as , 1/2 A THt ( y = A TH ( V " A THr { V " A THn ( V ( 31 ) AIL tliat remains is to compute the normal and radial thrust accelerations A TIIn anc * A THr at Rie re< 5 uire< ^ <- irn e points. The present radial components of thrust acceleration have already been given as A THr a; + B r A TH - (-p/R 2 + u> 2 R) (32) and at cutoff as A_ ir . - A' + THrf r B - A THf ” + w f B f ) (33) where the subscript f designates final values at cutoff. The centrifugal acceleration terms in Eqs. (32) and (33) can be rewritten, respectively, as 2 tjJ R = (34) ? 2 w*R f = M { Finally, by linearly interpolating the accelerations due to gravity and centrifu- gal force and by employing the value of acceleration magnitude derived pre- viously in Eq. (30), one obtains for A^,^^ at the required time points A THr (t j ) = A r + B r A TH (t j } " <^ /r2 + m2 / r3 ) - (j - l){[(-p/R 2 + M 2 /R f 3 ) - (-p/R 2 + M 2 /R 3 )]/4| (35) 1, ... 5 - 24 - The normal component of thrust acceleration is given simply as A THn ( V = A n + B n A TH ( V J = 1 > ■■■ 5 (36) By combining the results of Eqs. (29) and (31), one obtains the integrand of Eq. (24) at the specified time points M(t.) = R(t.)A THt (t.) (37) and, using Simpson's integration rule, one can compute an estimate of AM: AM' = [M(t } ) + 4M(t 2 ) + 2M(t 3 ) + 4M(t 4 ) + M(t 5 )]T'/l2 (38) where AM' is the angular momentum if the initial estimate of time to go is assumed. By comparing AM, the true angular momentum to be gained, with AM', one can compute an adjustment to the time to go T it T a = (AM - AM')/M(t 5 ) (39) Finally, the time to go becomes T g = T' + T g a (40) With time to go T and the time to mass depletion T , the acceleration O 1* integral V is computed as S V g Iv /(T j e ' rO t) dt = -V ln(l - T IT J e o' r 0 (41) - 25 - The guidance scheme described here is iterative. The steering coefficient calculations require an estimate of time to go T , while the time-to-go calculations presuppose a knowledge of the steering coefficients. Fortu- nately, experience has shown that with reasonable startup values and appropriate updating between major guidance cycles, the equations described here ccnverge quite rapidly for the mission considered. 5. INTEGRAL CONTROL In Paragraphs III. B. 2 and III. B. 3, expression's were derived for desired radial and normal thrust accelerations. If these accelerations were translated directly into vehicle axis pointing-direction commands, a thrust acceleration pointing e**ror would result because of vehicle ard IMU prop- erties, such as engine misalignments, vehicle center of gravity effects, and IMU gimbal misalignments. Integral control, the method for overcoming these pointing errors, is used to measure differences between desired and actual guidance- computed accelerations. These differences are numerically time-integrated with an appropriate weighting factor for stability reasons, i.e. , a digital filter, and finally subtracted from the desired radial and normal accelerations. The final steady state effect is to bias the vehicle- pointing commands so that the observed misalignments are canceled. Preference 7 contains a complete description of integral control. The equations are mechanized as AA ,r AA n + (C: F6 } - A n J l4Z) AA r -AA r + (c F6 ) [(AV r /At) - Aj (43) The arrow indicates that the results are stored in the original memory location . - 26 - where i i i > AA and AA n r integral control terms Cj.,^ integral weighting factor A^ - desired normal component of total vehicle acceleration A - desired radial component of total vehicle acceleration AV /At - guidance-computed normal component of total vehicle acceleration AV /At guidance - computed radial component of total vehicle acceleration At 1 -second major cycle interval The corrected desired normal and radial equations are then given as nc = A n kA (44) A rc 2 u> r (45) Equation (45) includes terms for gravity and centrifugal accelerations, these terms were added so that A^, excluding the effect of AA f) would represent the desired radial thrust acceleration. Dividing Eqs. (44) and (45) by thrust acceleration magnitude yields the desired unit vehicle longitudinal axis in the radial, normal, and tangential coordinates ID = A /A_„ &n nc TH (46) ID = A / A u &r rc TH (47) 1/2 U £t * (* - °*n - °fr) (48) - 27 - Then the unit desired vehicle longitudinal roll axis vector U| is converted to inertial coordinates by a matrix multiplication as fix" i C in> r-f- 1 - N U £n U- - U £r_ ( 49 ) wiiere N is the matrix that relates tangential, normal, and radial coordinates to EC1 (x, y, coordinates. The desired unit vehicle pitch axis is arbitrary from a guidance standpoint and is selected by the guidance equations to conform to vehicle and/or telemetry antenna constraints. Finally, the desired unit yaw axis is formed from U^ x to complete a right-hand orthogonal coordinate system. 6. MANEUVERING EQUATIONS This section presents a derivation of the maneuvering equations. In this paper, maneuvering is considered the application of rate limits to the guidance commands (desired vehicle axes) and, if the desired maneuver is large, the computation of an efficient maneuver. In addition, the maneuver- ing equations calculate values to interpolate between successive major cycle commands on a minor cycle basis. The attitude error equations, performed on a minor cycle basis, take the output commands from the maneuvering equations in the form of desired vehicle axis vectors. The desired vehicle axes are compared with the actual present vehicle axes (read from the 1MU attitude sensors), and vehicle attitude errors are computed. The flow is from guidance equations to maneuvering equations to attitude error equations. - 28 - The initial calculations made by the maneuvering equations are a coordinate conversion from earth centered inertial (ECI) to drifted launch- centered inertial (LCI), as follows: [rg] t = [0] T [*][cg] u £r - [RG] [U $ ] 5 6 (50) U^ = [RG] [U c ] where U^ c , U^, and U^ c are the final desired commanded- vehicle roll, pitch, and yaw axes in drifted LCI coordinates: [CG] is a coordinate trans- formation matrix from ECI to LCI coordinates; [p] is the present compen- sable IMU drift matrix; and 4* is a matrix chosen to rearrange the rows of <p so as to be compatible with the LCI coordinate system. The drifted LCI coordinate system^ is used throughout the following equations, since it is the most convenient coordinate system for the guidance minor cycle calculations; the IMU attitude readouts are referenced to the drifted LCI coordinate system. The drifted LCI system (called a, b, g) can also be considered an ideal gimbal angle system. Following the coordinate transformation, one can calculate the transformation matrix of the desired maneuver: u £c ' u 4a U !c- V u 4c ‘ u r M = ^nc ' °4a “V °na °nc- U„ 4a _V ' u u V' V U y 4a where Uc , U , and U r are the present desired commanded- vehicle roll, pitch, and yaw axes. The trace of the BG matrix can be related to the commanded maneuver as - 29 - Trace - 1 - 2 cos 8 (52) o r cos 0 (1 - Trace)/2 where 0 is the magnitude of the maneuver. Cos 0 can be tested to determine whether to command a rate- limited maneuver. If cos 0 is sufficiently close to 1.0, the maneuver is small and the following calculations are performed: dU !L = U ^c ' U U and (53) ciU T = U - U ol 0c na The variables dU^ and dU^j represent the motion of the commanded-body axes Ut and U during the next major cycle. If dU f . T and dU T are multi- plied by the reciprocal of the number of minor cycles per major cycle, C^q, the commanded-body axes can be interpolated on a minor cycle basis. With 6R = dUt T cm oP = dU T cm nL (54) computed on a major cycle basis, the following minor cycle calculations yield the interpolated axes: - 30 - cm ( 55 ) U |dm ^>ldm «- u <- u ^dm qdm + 6 R + 6P cm If cos 0 is not close to 1.0, a large maneuver has been commanded and rate- limiting is desired. The maneuvering strategy selected consists of maneuvering about a single inertially fixed axis at a constant angular rate. It can be shown (Euler's theorem) that this single-rotation axis is an eigenvector of the maneuver transformation matrix given in Eq. (51). The following paragraphs describe an algorithm for computing a command “eigenvector" maneuver. To solve for an eigenvector, E, of BG, the maneuver transforma- tion matrix, consider a skew- symmetric matrix D, calculated as D = BG - BG T (56) The D matrix has the properties [D] [E] = to] (57) D. . !J = 0 if i = j (58) From these properties of the D matrix, the following set of equations is derived: D 12 E 2 + D 13 E 3 " 0 ' D 1 2 E 1 + D 23 E 3 = 0 -°13 E 1 ' D 23 E 2 " 0 (59) - 31 - Solving for the eigenvector in Eqs. (59) yields E 1 ' D 23 BG 23 ' BG 32 E 2 ” " D 1 3 " BG 3 1 BG 1 3 (60) E 3 ^ D 1 2 = BG 1 2 ' BG 21 Alter unitization, the eigenvector is transformed from body coordinates to the LCI coordinate system with the following matrix multiplication: RL = [A] UE. UE. UE 3 J (61) where U UJ 1 the body axes as columns. The magnitude of the eigenvector computed in Eq. (60) is propor- tional to the sine of the maneuver angle 0. There is a singularity wherein the magnitude of the eigenvector approaches zero when the commanded maneuver approaches 180 deg. When cos 0 is sufficiently close to -1.0, the RL vector is replaced by a unit- commanded body axis according to the following strategy. The largest diagonal element of the BG matrix is determined. - 32 - Then, if BG largest, R L. = IL 1 1 BG,_ largest, RL _ U U L* M < BG,, largest, lOl = U 33 t,a The above procedure ensures that the initial maneuver away from the 180-deg region is not made about an axis which is perpendicular to the desired maneuver axis. RL is considered the unit command rotation vector, and the variables dU^^ and dU^ are computed as dU f j = (RL X U^ a ) MLIM dU = (RLX U ) MLIM PL ' Pa/ and the interpolated body axes, on a minor cycle basis, are computed as described in Eqs. (54) and (55). MLIM is the magnitude of the rate limit. The method used to interpolate the commanded-body vector throughout the major cycle results in a very small nonorthonormality in the commanded vehicle axes. To prevent this nonorthonormality from growing, one can per- form an orthonormalizing process for each major cycle, as follows: Lu = + dTJ ei. )/llJ u + d LiJ L,a =1 ha X % a *«O nL ,/|U 5a X (U, a + dU nL )l (64) «na = % X V This process resets the command- vehicle coordinates to orthonormality every major cycle. - 33 - c. NAVIGATION The- guidance navigation calculations compute the 'present vehicle position and velocity using as inputs accumulated incremental velocity pulses from a triad of force rebalance integrating accelerometers. One of the accelerometers (WC1 is mounted on the IV U stable platform (turret); the other two (l;C and VC) are mounted on a platform that rotates, or carousels, at I rpm. Figure 4 >s a block diagram of the major and minor cycle navigation . 1. ACCEL fill O V 1 1 T E R RESOLUTION AND COMPENSATION For the navigation function to he performed, the initial azimuth (at "go inertial") of the rotating platform must be determined. Further, whenever the incremental velocities are sampled, this azimuth must be updated. Four coordinate systems are used in the accelerometer resolution and compensation: [uc, vc, wcj [l,2,3] [u, v,w] [x,y, z ] Actual ca rouselling accelerometer input axes coordinate system. The coordinate system is, in general, nonorthogonal and rotating due to 1MU drift and ca rouselling. Ideal carouselling coordinate system This coordi_- nate system of convenience is defined as follows. 3 is colinear with the carouselling axis of rotation. 2 is perpendicular to 3; it is in_the plane of 3 and the vc accelerometer input axis. 1 completes the orthogonal right-hand set. Drifted launch site coordinate system. Initially equal to the initial vehicle roll, pitch, yaw coordi- nate system. After go inertial, related to the initial vehicle coordinate system by the IMU drift matrix. Earth centered inertial coordinate system. Defined as follows: x = unit vector in the equatorial and prime meridian planes at go-inertial time. z = unit earth spin vector, i.e., North Pole y - z x x to complete a right-hand set. 34 Figure 4. Navigation Block Diagram An miii.il > 1 /. 1 ‘nuih, or phase angle, is computed in the ground-in- flight interface program as follows: FANG - HTHYN + CFANG + CZZANG ( 65 ) where HTHYN = the angle, computed in the ground alignment program, from north to the VC accelerometer input axis plus gyro mis- alignment errors CZZANG = gyro misalignment IMU compensation parameter CFANG = parameter to convert the phase angle from northeast to dov/nrange-crossrange coordinates and to time synchronize with the accelerometer readouts. The sine and cosine of the initial phase angle are then taken for resolution purposes . SS = sin(FANG) BS = cos (FANG) ( 66 ) After initialization, the carousel phase angle sine and cosine is updated every 40 msec by a trigonometric identity as follows: SS n+l = < SS n Hcos 0.24°) + (BS )(sin 0.24°) BS n+1 = (BS n )(cos 0.24°) - (SSJisin 0.24°) (67) where 0.24 deg is the angle : ’car luselled" through in 40 msec. Also, the carousel phase angle is reinitiaJ zed to SS and BS at 1 -minute intervals. The accelerometer incremental counts are modified with the calibra- tion constants determined as part of factory calibration and pad test procedure The 40-msec frequency accelerometer compensation equations are AV uc = (CKAX)(AN - CKBXG) + (CKNX)(AN ) 2 uc A ^vc (CKAY)(AN vc - CKBYG) + (CKNY)(AN v( J 2 AV wc = (HZAF )(AN - HBZ) + (CKNZ)(AN ) 2 wc whe re AN uc’ AN vc’ AN wc the raw counts from the carouselling uc and vc and the stationary wc accelerometers over the last 40-msec interval CKBXG, CKBYG = the uc and vc accelerometer factory bias calibration HBZ - ^e wc accelerometer bias calibration computed during final align CKAX, CKAY = the uc and vc factory scale factor calibration HZAF = the wc accelerometer scale factor calibration computed during final align CKNX, CKNY, CKNZ = the uc, vc, and wc nonlinearity calibration Continuing, one finds that ST1 = AV uc " ( czzb 2U)(AV ) - (CZZB3U)(A V t vc wc’ ST2 = AV - (CZZB3V)(AV ) VC WC ' (69) Equation (69) shows small angle approximations to misalignment rotation compensation where ST1 , ST2 = velocity increments in 1,1,3 coordinates CZZB2U - misalignment of the u axis toward ~Z CZZB3U = misalignment of the u axis toward 3 CZZB3V = misalignment of tne v axis toward 3 - 37 - ST1 and ST2 are then resolved into u,v,w coordinates as follows: IBl . mm AV um = (BS HST1) - (SS)(ST2) AV vm = (SS)(ST1) + (BS)(ST2) The resolved minor cycle velocity increments are finally accumulated as follows: A V <- AV A V up up um AV AV + AV vp vp vm [T J I ■ *>W'! * A V f AV + A V wap wap wc Once per second, the accumulated velocity increments are sampled for use as inputs to the major cycle navigation equations AV «- AV u up AV *- AV V vp AV «- AV wa wap and AV are chei up vp wap further accumulation in the next major cycle interval. AV 0 up AV «- 0 vp AV <- C wap The w-accelerometer misalignments are performed on a major cycle basis, since the w instruments do not carousel. These misalignments do rotate, because the w-accelerometer is rotating at earth rate at liftoff. During guidance initialization, misalignments are computed in the u, v, w coordinate system as a function of the initial gimbal angle bWl = (CKB4W)(sin aj + (CKB5W)(cos a) BW2 = - (CKB4W)(cos a.) + (CKB5W)(sin a.) where BWl - w-accelerometer misalignment away from u axis BW2 - w-accelercmeter misalignment away from v axis CKB4W, CKB5W = factory-calibrated w-accelerometer misalignments in an a gimbal-oriented coordinate system The w-accelerometer input axis also "cones" after go inertial as a function of the inner gimbal angle, because of misalignment of the inner gimbal axis and the carouselling axis. The complete equation for w-accelerometer misalignment compensation is given in Eq. (75): AV = A V + [BWl + (CKBSW)(cos a) - (CKBCW)(sin a)]AV w w & u. + [BW2 - (CKBCWMcos a) - (CKBSW)(sin »)]AV CKBSW, CKBCW = calibrated carouselling axis misalignments in an a gimbal angle coordinate system - 39 - 2 . GYRO DRIFT COMPENSATION Gyro drift compensation is accomplished by computing the platform drift rate with calibration values obtained during hangar and pad tests. Then by time -integration of the drift rate, a matrix p can be computed that trans- forms the present (u,v,w) axes to the initial nondrifting axes. Further matrix operations then transform to the desired ECI coordinate system. rAV l rAV -i sx u AV sy = [CG] r [4>] T [p] AV V AV L sz J AV L w_ ( 76 ) where AV sx, sy, sz measured (sensed) values of velocity increments in the ECI coordinate system AV u, v, w measured value of velocity increments along the, present drifted launch site coordinate system p - d rift matrix 4* row rearrangement matrix CG - coordinate transformation matrix from LCI to ECI coordinates Note that p, cp and CG are shared by guidance (Paragraph III.B.6) and navigation. If the instantaneous drift rate vector is given as (p u> p v , p w ), the derivative of each column of the p matrix is the cross-product of the drift rate vector with that column vector. Calculating the cross products and combining terms, one obtains the matrix form p = [p] 0 6 \ w -p 0 < P , w -p u 0 ( 77 ) Integration of p is accomplished by using a second order Runge- Kutta algorithm. First, the p matrix is integrated a half cycle with an initial derivative: K~~ *n + < At/2) * n (78) The drift rate is then averaged over two cycles: A p = (A p + A p , , ) / 2 r u r un r un+l A p = (A p + Ap , . ) / 2 ( < ^) r v r vn r vn+l' A? = (A p + A 0 , , )/2 T w r wn r wn+l' A final step advances p as follows: n+1 1 P n + [p 1 L V n , 0 A ^ -Ap r w Ap r w 0 -A? u -Ap r v Ap y u 0 (80) Computation of the drift vector ip in u, v,w coordinates is complicated by the rotation of the carouselling instruments. Drifts of each gyro must be located spatially due to the arbitrary location of the 1MU turret and the rotation of the platform at 1 rpm. All drift matrix calculations are performed once per second. A central assumption for the drift calculations is that the drift occurred approxi- mately centered in the 1 sec compute cycle. To locate the carouselling instruments, one must compute backed-up values of the variables BS and SS. SSB = -(BS)(sin 3°) + (SS)(cos 3°) BSD - (BS)(cos 3°) + (SS){sm 3°) ( 81 ) The variables SSB and BSB are then used to resolve AV and AV into u v coordinates representing the "average" position of the carouselling gyros over the last second. AVgi = (AV u )(BSB) + (AV v )(SSB) AV g2 = -(A~V u )(SSB) + (AV v )(BSB) (82) The fixed-torque and unbalance drifts of the carouselling gyros are then computed as DRFTU FT DU + (CKU1)(AV t ) + (CKU2)(AV g2 ) DRFTV = FTDV + (CKU3)(AV g2 ) + (CKU4)(AV gJ ) (83) where FTDU, V = fixed-torque drifts CKU 1 , 3 = spin-axis unbalance drifts CKU2, 4 = input-axis unbalance drifts In addition to the usual error sources associated with an IMU, carouselling itself introduces gyro drifts. The compensable ones are called gimbal- oriented bias (GOB), turret-oriented bias (TOB), and turret-oriented eta (TOE). GOB is a drift in the carouselling plane, which rotates as the inner gimbal rotates; i. e. , a drift fixed to the inner gimbal. Initially, GOB is computed in launch-centered coordinates as - 42 - GOBC1 = (CK20)(cos CK21) GCBC2 = (CK20)(sin CK2.1) ( 84 ) where CK20 - GOB magnitude CK21 = GOB phase angle 7 Later, in flight, the computation of GOB reflects the rotation about the inner gimbal as follows: GOBI - - (GOB C2)(cos a) + (GOB Cl)(sin a) GOB2 = (GOB Cl )(cos a) + (GOB C2)(sin a) (85) I urret- or tented bias is a drift of the tui ret (perpendicular to the carouselling plane) that is a function of several harmonics of the turret present position with respect to the inner gimbal. To locate the turret during guidance initialization, one must compute the angle of the turret revolution from a zero inner gimbal angle value: - tan (sin Q-. /cos a. ) where °' 1 - initi al (at go inertial) inner gimbal angle sin a. = sine of initial a cos or. = cosine of initial a - 43 - Then, in flight, this a,ngle is updated as - tan (sin a ! cos a) + ( 87 ) The equation for TOB is 4 TOB = [cos (n a - 0 ) f— { i L 'it l i=i ( 88 ) In Eq. (88), n. represents integers specifying the harmonics of a ^ to be included in the TOB calculation,, 6^ represents the phase angle of each harmonic; and K represents magnitudes. TOE is a drift in the carouselling plane that is a function of the location of the turret with respect to the launch- centered coordinate system. It is computed at guidance initialization as TOE 1 = - (CK22)[ (sin a.)(cos CK23) + (cos *.)(sin CK23) J TOE2 = (CK22)l(cos m)(cos CK23) - (sin a.)(sin CK23) ] (89) who re CK22 TOE bias drift magnitude CK23 TOE bias drift phase angle Finally, the derivative of <p is computed as shown in Eq. (90): A p u = (BSBKDRFTU) - (SSB)(DRFTV) + GOBI + TOE1 Ap v = (SSBMDRFTU) + (BSB)(DRFTV) + GOB2 + TOE2 (90) A p w = HRZ + (DIAU)(AV ) - (CZZU6)[ (AV v )(sin a.) + (AVJfcos ».)] + TOB - 44 - V whe "e 3. BSB, SSB = sine and cosine of the average carousel angle during last compute cycle DRFTU, DRFTV = fixed-torque and g-sensitive drifts of the UC and VC carouselling gyros GOBI, GOB 2 = GOB drift [Eq. (84)] TOE 1 , TOE 2 = TOE drift [Eq. (89)] TOB = TOB drift [Eq. (88)] HRZ = noncarouselling Z gyro fixed-torque drift DIAU - spin-axis unbalance drift CZZU6 = input-axis unbalance drift, Z gyro TRANSFORMATION OF VELOCITY TO INERTIAL x,y,z COORDINATE SYSTEM The inertial x, y,z coordinate system is defined as follows: x = unit vector in the equatorial and prime meridian planes at go inertial time z = unit earth spin vector, i. e. , North Pole y = z X x to complete a right-hand set 4. VEHICLE INERTIAL POSITION The following trapezoidal integration formulas are used to obtain the vehicle position (X, Y, Z) in the inertial coordinate system: X <■ X + V (At) + (1/2) (At) 2 (AV + AV ) Y «- Y + V (At) + (1/2) (At) 2 (AV + AV ) (9D y y o y Z Z + V (At) + (1/2) (At) 2 (AV + AV ) Z b Z gZ - 45 . where V , V , V x y z At AV , A V sx AV , A V g x sy gy » A V sz ,AV gz x, y, z components of vehicle velocity major compute cycle interval - 1 second x,y,z components of sensed vehicle velocity increment from Eq. (76) x, y, z components of velocity increments due to gravity Vehicle position magnitude is then calculated as 2 2 2 1/2 r = (x + y + z ) (92) 5. GRAVITY COMPUTATIONS An approximation for the gravitational potential of the earth is U U = (n/R)[l + (Ja 2 /3R 2 )(1 - 3 sin 2 \) ] (93) where p = gravitational parameter of the earth R = distance from center of the earth \ - latitude a = equatorial radius of the earth J = a characteristic constant that is a function of the moments of inertia with respect to the polar and equatorial axes A derivation of Eq. (93) is contained in Ref. 7. This approximation, which is a truncated series, contributes negligible inaccuracies to the overall navigation function. - 46 - Then, from the definition of the Z component of vehicle position, it follows that sin 2 \ = 2 2 /R 2 (94) which, upon substitution into Eq. (93), yields U = (p/R) (1 + Ja ? '/3R 2 - Ja 2 Z 2 /R 4 ) (95) The x, y, z components of the acceleration due to gravity are then obtained by the partial differentiation of Eq. (95) with respect to each axis, and g x = -aU/9x = -(8U/9R)OR/ax) gy = -au/a y = (-au/aR)OR/a y ) (96) g = -au/az = (-au/aR)OR/az) The negative sign is added by convention. An evaluation of the common term 3U/aR yields aU/aR = (-p/R 2 )(l + Ja 2 /R 2 - 5Ja 2 Z 2 /R 4 ) (97) 2 and, by factoring out - -(p/R ), one obtains aU/3R = A (1 - Ja 2 A /p + 5 Jr. 2 A Z 2 /pR 2 ) (98) 8 8 8 - 47 - In addition, one finds that dR/dx = X/R 9R/9y = Y/R dR/dx = Z/R (99) and aU/az = ZfxJa i 2 Z/R 5 = +2Ja Z A 2 Z/|j.R 8 ( 100 ) The velocity increments over a major compute cycle due to gravi- tational acceleration arc given in Eq. (101). AV gx g x -(ou/aR)(aR/ax) a v g - -(au/aR)(aR/ax) Bi y AV =g - -OU/9R)OR/8z) gz b z 6. VEHICLE INERTIAL VELOCITY The inertial velocity of the vehicle is computed as ( 101 ) (V .). (V) + (1 /2)[(AV .) , + (AV .) ] [At] + (AV )(At) in) 1 in L gi n+1 gi n J 1 J si i = x,y, z gi ( 102 ) Averaging AV over two cycles produces a trapezoidal integration of acceler- D. ATTITUDE ERRORS Attitude errors art? < rucial outputs of the guidance equations utilized by the Digital Flight Controls Equation in stabilizing the vehicle to a desired attitude. They represent differences between the present actual vehicle attitude and the present desired vehicle attitude. The desired attitude is computed by the guidance maneuvering equations (Paragraph III. B. 6); the actual attitude is derived from synchros attached to each gimbal. 1 . form SYNCHROS The synchro signals supplied to the inflight computer are of the VI = K sin (0-120°) and V2 = K sin (0-60°) (103) where G can be any of three gimbal angles, a, p, or y„, and K is a scaling factor common to VI and V2. By trigonometric identities, one obtains VI - V2 = K sin (0-120°) - K sin (8-60°) = K[(sin 0 cos 120° - cos 0 sin 120°) - (sin 6 cos 60° - cos 0 sin 60°)] = K sin 0 (cos 120° - cos 60°) (104) -K sin 8 - 49 - a n<l VI I V2 K sin (0 - 120°) + K sin (0 - 60°) K | (sin 0 cos 120° - cos 0 sin 120°) + (sin 0 cos 60° - cos 0 sin 60°) | (105) - K cos 0 (- sin 120° - sin 60°) = -K*/3 cos 0 Thus, by simple sum and difference, one obtains the sine and cosine of the gimbal angles, except for a common multiplier -K and the constant V 3. Since the sine and cosine of the gimbal angles are desired on a minor cycle basis as a convenience for the computation of vehicle attitude errors, it is necessary to determine the scaling factor K. At go inertial, the following initial calibration is performed for each of the three gimbal angles: X - V2 - VI Y = (VI + V2)(C1) (106) K 2 - (X)(X) + ( Y)( Y) K -v/ K 2 In this series of equations. Cl is a constant equal to -1.0/\f3, and K is the calibrated value of the synchro scale factor. On succeeding minor cycles, the sine and cosine of each gimbal angle are computed as follows: - 50 - X = V2 - VI m Y m = < V1 + V2 > (C1) K 2 - (X )(X ) + (Y )(Y ) m m m m m K - (1/2) fK 2 /K + K ) m m m rn 107 ) sin 0 = X/K cos 0 = Y/K m m These minor cycle computations use Newton’s square-root algorithms, which converge quite rapidly if the initial value is relatively accurate. Since the three gimbals, n, (3, and y^, are not necessarily zero at go inertial, they are read initially; the initial values are used to compute the three angles through which the vehicle rotates during the flight. Taking a as an example, trigonometric identities yield sin a = sm a cos a. - cos a sin a pi pi cos a = cos a cos a. + sin a sin a pi pi (108) where sin a, cos a = sine and cosine of vehicle angular rotation sin a , cos a = present gimbal synchro sine and cosine readings sin o , cos m = initial gimbal synchro sine and cosine readings An identical procedure is followed to compute the sine and cosine of the }3 and v D vehicle rotations. - 51 - The fourth gimbal y has very limited travel (‘"10 deg), and the following simplifying approximations are made: cos y - 1.0 ( 109 ) sin y ~ (CKDSCA}(sin 'y ) - sin y^ where CKDSCA = calibrated scale factor for sin ATTITUDE ERROR COMPUTATIONS The IMU gimbal angles (a, p, Y and Y^) ideally establish the relationship between two vehicle coordinate systems: the present roll, pitch, and yaw (£,q,p body axes; and the initial body axes (£.,C,£.l. Initial alignment is such that, at launch, a positive roll results in a negative gimbal angle «; a positive yaw results in a positive gimbal angle (3; and a positive pit< h results in a positive gimbal angle Y^. The fourth gimbal Y, in normal circumstances, is zeroed. Its sign convention is the same as Y i\. [uc.r,|.u, CO* tp 0 -sin 0 I 0 \ K <» * o« \ (' sin P o i » ■<*> > i - sin ,1 ( ,1 1(1 (j I I ljr , 0 .] (110) ij si n it i 06 I 06 / | Performing the matrix multiplication with cos y ~ 1.0 yields - 52 - (' '>■> 1 1 ( , ‘ " i“i ) l( --'ii 'i ) (» ■ >& \ ^ sin 6 cos u ( - cos ^ ^ sin >i sin / - cos cos 0 sin \ sin a - cos y^ c ob 3 sm \ cos a - -nn ^ sin o) - sin Yp cos <>) (- bin |i) (t os 3 cos o + ( - cos (1 sin a + sin 3 sin \ tos sir* 3 sm y s in a (sin > ^ cos il t cos y ^ sin y (sin Y{^ sin 3 cos a (- sin y ^ sin 3 sin u - sin cos p sin y sin o - 6in y^ cos 3 sin y cos a + cos sin ») f cos y . Cos o) :n i) or M = lUg, U^, U^j If the matrix of initial vehicle axes is assumed to be identity, the rows of the M matrix comprise the present measured vehicle axes in a gimbal-oriented coordinate system (a,b,g), where initially a = Uc b = U ni ( 112 ) From the actual vehicle axes and the present desired vehicle axes, attitude errors can now be computed. Consider a three-axis (roll, pitch, yaw) maneuver to reorient the vehicle from its actual attitude to the desired attitude, as three Euler rotations in the following order: roll, pitch, yaw. In matrix form, tins is - 53 - 0 0 B O cos R sin R cos P 0 -sin P I f cos Y sin Y 0 ~J -sin Y cos Y 0 0 - sin R cos R sin P 0 cos P where R, P, and Y are the respective roll, pitch, and yaw rotations. The matrix multiplication yields (113) cos P cos Y -cos P sm Y sin P sm R sin P cos Y + cos R sm Y sm R sin P sin Y + cos R cos Y -sin R cos P •cos R sin P cos Y + sm R sin Y cos R sin P sin Y + sm R cos Y cos R cos P (114) Simplifying B with small angle approximations yields "1 Y -P B« -Y 1 R P -R 1 (115) Another expression for B in terms of body axes is §dm - r, 5 ^ £dm • U n Ut . §dm • U £ > B - 0 . pdri) '°£ 0 . i|dm • U n U . r)dm (116) %m f,dm • U n i t,dm • D J REFERENCES 1 E. Da maud , D. L. Kleinbub, and J. B. Shaul, Guidance, Control and Ground Equations for SSLV C-26 , Aerospace Corporation Report No. TOR- 01 7 2(21 { 2- 02)- 1 1 , Reissue B (1 November 1972), Vol. I. Program 624A Mission Specification for Flight Plan VII- J , Aerospace Corporation Report No. TOR-01 72(21 1 2- 02)-6 , Rev. 1 (28 September 1972). H. Sokoloff, Program 624A Discrete List for Flight Plan VII- J , Aerospace Corporation Report No. TOR-0172(21 12-02)-l6, Reissue A (24 November 1972). George W. Cherry, "A General Explicit Optimizing Guidance Law for Rocket Propelled Spacecraft, 11 Proceedings of the AIAA/ION Astrodynamics Guidance and Control Conference , AIAA Paper No. 64-638 (24-26 August 1964). D. MacPhcrson, An Explicit Solution to the Powered Flight Dynamics of a Rocket Vehicle , Aerospace Corporation Report No. TOR- 1 69(31 26)TN-2 (31 October 1962). C. W. Pittman, The Design of Explicit Guidance Equations for Rocket Ascent, Aerospace Corportation Report No. T DR- 469(5 5 40 - 1 0)- 4 (24 May 1965). F.E. Darnaud, A Technique Allowing Continuous Operation of Integral Control , Aerospace Corporation Report No. TO R- 1 00 1 (2 1 1 6 -60)- 8 (1 3 September 1966). K. A. Ehricke, "Environmental and Celestial Mechanics," Principles of Guided Missile Design, Vol. I: Space Flight (D. Van Nostrand Company, Inc., Princeton, N. J. , I960). Carousel V Inertial Navigation System, System Technical Description EP0137, Delco Electronics Division of General Motors, Milwaukee, Wisconsin, 1969. - 57 - PRECPDING PAGE BLANK-HOT FI